Front Page Titles (by Subject) Confusion between Mathematical and Exact Sciences. - The Theory of Political Economy
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Confusion between Mathematical and Exact Sciences. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Confusion between Mathematical and Exact Sciences.
Many persons entertain a prejudice against mathematical language, arising out of a confusion between the ideas of a mathematical science and an exact science. They think that we must not pretend to calculate unless we have the precise data which will enable us to obtain a precise answer to our calculations; but, in reality, there is no such thing as an exact science, except in a comparative sense. Astronomy is more exact than other sciences, because the position of a planet or star admits of close measurement; but, if we examine the methods of physical astronomy, we find that they are all approximate. Every solution involves hypotheses which are not really true: as, for instance, that the earth is a smooth, homogeneous spheroid. Even the apparently simpler problems in statics or dynamics are only hypothetical approximations to the truth.1
We can calculate the effect of a crowbar, provided it be perfectly inflexible and have a perfectly hard fulcrum,—which is never the case.2 The data are almost wholly deficient for the complete solution of any one problem in natural science. Had physicists waited until their data were perfectly precise before they brought in the aid of mathematics, we should have still been in the age of science which terminated at the time of Galileo.
When we examine the less precise physical sciences, we find that physicists are, of all men, most bold in developing their mathematical theories in advance of their data. Let any one who doubts this examine Airy's "Theory of the Tides," as given in the Encyclopædia Metropolitana; he will there find a wonderfully complex mathematical theory which is confessed by its author to be incapable of exact or even approximate application, because the results of the various and often unknown contours of the seas do not admit of numerical verification. In this and many other cases we have mathematical theory without the data requisite for precise calculation.
The greater or less accuracy attainable in a mathematical science is a matter of accident, and does not affect the fundamental character of the science. There can be but two classes of sciences—those which are simply logical, and those which, besides being logical, are also mathematical. If there be any science which determines merely whether a thing be or be not—whether an event will happen, or will not happen—it must be a purely logical science; but if the thing may be greater or less, or the event may happen sooner or later, nearer or farther, then quantitative notions enter, and the science must be mathematical in nature, by whatever name we call it.
This subject of the approximate character of quantitative science is pursued, at some length, in my Principles of Science, chap. xxi., on "The Theory of Approximation," and elsewhere in the same work.
[]Thomson and Tait's Treatise on Natural Philosophy, vol. i. p. 337.